STUDY GUIDE FOR PROBLEM # 20

Dania Elder

Problem 20 :

How many 5-digit ZIP codes numbers are possible if consecutive digits must be different?

  • First by¬†reading the problem you see there are 5 possible spaces to fill¬†¬†¬†¬†¬†¬†¬†¬†¬†¬†¬†¬†¬†¬†¬†¬†¬†¬†¬† _¬†_ _ _ _
  • Next you see that consecutive digits must be different. Therefore you cannot have a zip code such as 11234, because the two 1’s are consecutive with one another.
  • So for the first spot you have 10 possible outcomes (0-9) as a zip code possibility.¬† 10 _ _ _ _
  • The next spot cannot be the same number as the first spot so there are 9 possible out comes

10 9 _ _ _

  • And the same for the next three spaces.

10 9 9 9 9

  • Lastly, you Multiply¬†the possible outcomes.

10*9*9*9*9 = 65,610 

Study Guide For Problem # 6

Ramsuresh Rampersad

I’m working on problem # 6

6.) A box contains 30 marbles: 15 red, 10 blue, and 5 green.

a) two marbles are selected with replacement. Find the probability that both marbles are red.

b) Two marbles are selected without replacement. Find the probability of first selecting a blue then a green marble.

SOLVING PART (a)

To solve part (a) of this problem first considered the following:

1. There is a total of 30 marbles in the box.

2. There are 15 red marbles.

3. The marbles that are chosen are being replaced.

If you keep these three ideas together, it will help you to avoid confusing the different color¬† marbles and second, that you don’t keep the marbles, they are being replaced in the box. Alright! Having said that, let’s begin solving this really simple problem. Don’t worry, we’ll go step by step. And as you follow these simple steps, which I am absolutely sure you can, you will soon discover that this problem is a walk in the park and you can solve this and others just like it.

Step 1: As we discovered earlier when we took those three things into consideration. There are 15 red marbles in the box. Now, we also know that we are going to select two marbles. But first we have to find the probability of selecting the first red marble. Doing this is really easy. Here’s how it’s done. We both know that there are 15 red marbles in the box, and that there is a total of 30 marbles in that box. So we dip in take a marble out (no peeking please! lol). What are the chances that you have just pulled out a red marble? Well that’s easy. Since there are 15 red marbles and a total of 30 red marbles. The chances or probability that you just pulled out a red marble must be (you said it!) 15 out of 30. So the probability that your first marble is red is 15/30. Let’s make that into a smaller fraction 15/30 = 1/2.

Step 2: We’re doing good, told you it was simple! Hey you can’t take that marble with you. So now please replace that red marble in the box. So now since you’ve replaced that red marble we now have once again, that’s right 15 red marbles. I know you’re thinking it and you are absolutely right, our total number of marbles is still 30. So don’t forget, we have a total of 30 marbles in the bag, including 15 red marbles because you replaced the red that was initially taken out. Awesome!

Step 3:¬† Ok, we’re doing great. We are on the road to finding the probability of choosing a second red marble. And guess what we’re both right because, it is exactly the same as the first situation. We’ve replaced that red marble we took out. Therefore, the probability of choosing a second red marble is also 15/30 = 1/2. Doesn’t seem like a problem at all does it?

Step 4: I know you’re gonna be like what! When I say this, but yes you’ve successfully come to the last step of this half of the problem. This is so easy you’re gonna laugh and say damn why didn’t I choose this problem to do…lol. Guess what? Now all you have to do for your final answer is to multiply those two probabilities you just found above.That is: 1/2 * 1/2 = 1/4. And you have just successfully solved the first half of this problem.

Answer = 1/4

SOLVING PART (b)

To solve this part of the problem first consider the following:

1. There is a total of 30 marbles.

2. There are 10 blue marbles, and

3. There are 5 green marbles.

4 When a marble is removed it is not replaced.

Step 1:¬† This part is just as easy. Start by following the same principles that we did earlier. That is separate the information as¬† above. Kool, so let’s get started. We’re going to take two marbles out the box, and this time we’re looking for the chance or probability of taking out a blue marble first. Hey this time you can keep the marble, we’re not putting it back.

So let’s see, how many blue marbles do we have, yes,10 blue marbles, and as you know we have a total of 30 marbles in the box. OK, so go ahead dip in. So did you get a blue one? What’s¬† the probability of selecting the first blue marble? Well, since there are 10 blue marbles and there is a total of 30 marbles. That’s right, it is: 10/30. It’s that easy. And 10/30 can be simplified 10/30 = 1/3.

DO NOT FORGET WE’RE NOT REPLACING THE MARBLE IN THE BOX. WE’RE PLAYING FOR KEEPS THIS TIME! So how much marbles does that leave us in the box? We had 30, you took one, therefore, we have only 29 marbles left in the box and NOT 30. Alright I can live with that!

Step 2: You get another chance, remember we’re choosing 2 marbles out of the box. We took one out that you kept leaving us with 29. And now we’re going to take another one out. Now what are the chances or probability that it is a green one? We’ll that easy too. I think you know where I’m headed. Yes, since there are 5 green marbles, the probability of selecting the first green marble is: 5/29. It’s out of 29 because we didn’t replace the first one we took out you were allowed to keep it. If we reduce 5/29 you will get a mixed number so it’s best to leave it as it is.

Step 3 : Well, what do you know? We have successfully reached the last step of this half of problem #6. And again all we do in this step is we multiply both probabilities to find our answer of first selecting a blue and then a green marble: 1/3 * 5/29 = 5/87. And that’s it. You have just completed # 6. And thank you!

Answer = 5/87